Classical and quantum constraints

Abstract

Many systems involve constraints that generally lead to dynamical motion that is constrained in one way or another. For example, three-dimensional dynamical motion that is constrained to lie on a fixed two-dimensional surface, e.g., motion effectively lying on the surface of the Earth, would constitute constrained motion. Oftentimes, constraints are accompanied by superfluous degrees of freedom, variables whose values are not determined by the equations of motion but whose values must be chosen to fully determine the temporal evolution of the remaining variables. Such variables whose values are left undetermined by the equations of motion are called “gauge” variables; and the somewhat arbitrary choice of such variables to fix the behavior of all the variables is called “choosing a gauge.”